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Asymmetric oscillations during phase separation under continuous cooling: A simple model
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View: Figures


Image of FIG. 1.
FIG. 1.

(a) Part of the phase diagram for mixtures of methanol-hexane. The points depict the experimentally determined phase-transition temperatures, as determined from turbidity measurements. denotes the width of the biphasic region. (b) Snapshots of the observed oscillatory phase separation under continuous cooling at a rate of 10 K/h. Volume fraction of methanol . Images were taken after 1010 s , 1085 s , 1135 s , 1150 s , 1260 s , and 1385 s , from left to right, respectively. The white part indicates that droplets exist and the dark part indicates that the system is transparent. The system suddenly becomes turbid, and after a while it becomes transparent again due to droplet sedimentation. These processes are repeating. (c) Space-time plot of turbidity oscillations at 10 K/h from 36 to . It is obtained by assembling line data, which were integrated snapshots in horizontal direction. The total time is 3600 s.

Image of FIG. 2.
FIG. 2.

(a) The model system consists of two phases, in the upper and lower space, each of which is assumed to be homogeneous in the gravity direction. Each phase is described within a horizontal region (gray region). (b) Scheme of the temperature-composition behavior. The thick lines indicate the composition of the upper/lower phase . At the black circle, droplets are created in the upper phase, which grow and sediment later by moving to the lower phase. The dashed arrow indicates the shift of composition due to droplets exchange.

Image of FIG. 3.
FIG. 3.

Symmetric oscillations. (a) Evolution of the compositions . At the open circles, the upper/lower phase nucleates droplets. (b) Time evolution of the total droplet area. The upper and lower parts of the figure correspond to the turbidity of the upper and lower phase, respectively. (c) Time evolution of the volume of the lower phase. (d) Enlarged display of the first nucleation sequence shown in (b) of the upper phase at . The total composition , the initial temperature , the reduced cooling rate , the system size , the critical domain radius , and the parameter in the CH equation .

Image of FIG. 4.
FIG. 4.

Asymmetric oscillations. (a) Evolution of the compositions . (b) Time evolution of the total droplet area, which is a measure for the turbidity. (c) Time evolution of the volume of the lower phase. The total composition , while the other parameters are identical to those in Fig. 3.

Image of FIG. 5.
FIG. 5.

(a) Phase-diagram in the plane for systems for cooling rates in the range . The area under the solid line corresponds to the one-phase region. “A” indicates asymmetric oscillations for all investigated. “S” indicates symmetric oscillations for all . “X” indicates a region, where depending on , either both phases or only the majority phase oscillates. (b) Sketch of the phase separation mechanism as a function of the total composition . For , the turbidity of both phases oscillates. For , only the majority phase oscillates if the initial temperature is close to the binodal line.

Image of FIG. 6.
FIG. 6.

Space-time plots for the composition dependence of turbidity oscillations of a binary mixture methanol/hexane. The volume fractions of methanol were , , and . Before collecting data, the samples were prepared in the one phase region, cooled to the initial temperature , and kept for 1 h at this temperature. At the beginning, (a) and (b) are in the two-phase region, whereas (c) still remains in the one-phase region. The samples were cooled at a rate of starting from . Total time is 4000 s.

Image of FIG. 7.
FIG. 7.

The period of turbidity oscillations in the upper phase is plotted as a function of the cooling rate for a total composition and an initial temperature . The solid line corresponds to a system size , and the dashed line to . In the inset, the same data are shown in a log-log plot. (b) Evolution of the composition in the upper phase , for the same set of parameters as in (a), and . Solid line is for and dashed line is for .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Asymmetric oscillations during phase separation under continuous cooling: A simple model